A Company Operates A Drinking Water Purifying Process At Two Different Sites And There Is Interest In Examining The Puri 1 (108.42 KiB) Viewed 22 times
A company operates a drinking water purifying process at two different sites and there is interest in examining the purity of the water produced. Data have been collected from each site over n weeks. The data are denoted by {(x1, yı), ..., (Xn, Yn)}, where x¡ and yi denote a measurement of purity, on a scale between 0 and 1, at sites 1 and 2 respectively, in week i. An appropriate probability model is given by the density function f(x,y) = 38xB-178-1 where 0 < x,y< 1 and 3,8 > 1. The two parameters B and 8 determine the marginal distributions of purity at sites 1 and 2 respectively. If B and 8 are equal the purity in each site is the same. The joint likelihood function for B and 8 is: n L(8,8) «[[68x9-479-4 = gagn] - f***] ) β = (24/01) i=1 i=1 The joint log-likelihood function is 4(8,8) = nloge (B) + nloge (8) +(B – 1)loge(xi) + (8 – 1)Eloge yi) Your task is to test the following hypotheses of interest: H: B = 8 H:B 8 You are asked to do this by i) calculating a 95% Confidence Interval for ß-and ii) by carrying out a Generalised Likelihood Ratio Test (GLRT).
d) Use H to calculate a Wald 95% Confidence Interval for B - 8 and provide an interpretation of the interval. e) Calculate the maximised log-likelihood under Hz. f) Calculate the maximised log-likelihood under H. g) Calculate the number of restrictions for this test. h) Derive the rejection region based on the chi-square distribution (set the significance level a to be 0.05). i) Make a conclusion as to whether or not there is evidence of a significant difference in the purity of the water produced at the two sites.
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